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The probability of India winning a test match against Australia is $\frac{1}{2}$ assuming independence from match to match. The probability that in a match series India's second win occurs at third test match is
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Verified Answer
The correct answer is:
$\frac{1}{4}$
Required probability
$$
\begin{array}{l}
=\mathrm{P}(\text { First win }) \times \mathrm{P}(\text { First win }) \times \mathrm{P}(\text { Second } \\
\text { win) } \\
+\mathrm{P}(\text { First Defeat }) \times \mathrm{P}(\text { First win }) \times \mathrm{P}(\text { Second } \\
\text { win) } \\
=\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}+\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}=\frac{1}{4}
\end{array}
$$
$$
\begin{array}{l}
=\mathrm{P}(\text { First win }) \times \mathrm{P}(\text { First win }) \times \mathrm{P}(\text { Second } \\
\text { win) } \\
+\mathrm{P}(\text { First Defeat }) \times \mathrm{P}(\text { First win }) \times \mathrm{P}(\text { Second } \\
\text { win) } \\
=\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}+\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}=\frac{1}{4}
\end{array}
$$
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